A356133 -id:A356133 - cp's OEIS frontend

A354384 Difference sequence of A356133.

2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3

Offset: 1

Clark Kimberling, Aug 04 2022

Cf. A026430, A356133, A091855 (positions of 2), A036554 (positions of 3), A091855 (positions of 4).

u = Accumulate[1 + ThueMorse /@ Range[0, 200]]  (* A026430 *)
v = Complement[Range[Max[u]], u];  (* A356133 *)
Differences[v] (* A354384 *)

a(n) = A007413(n) + 1.

a(n) = A036580(n) + 2.

A026430 a(n) is the sum of first n terms of A001285 (Thue-Morse sequence).

Original entry on oeis.org

0, 1, 3, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 19, 21, 23, 24, 26, 27, 28, 30, 31, 33, 35, 36, 37, 39, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 57, 59, 60, 61, 63, 65, 66, 68, 69, 70, 72, 73, 75, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90, 91, 93

Offset: 0

Clark Kimberling

Winston de Greef, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.

Cf. A001285, A356133 (complement).

Cf. A115384.

a026430 n = a026430_list !! n
a026430_list = scanl (+) 0 a001285_list -- Reinhard Zumkeller, Jun 28 2013

A001285 = Table[ Mod[ Sum[ Mod[ Binomial[n, k], 2], {k, 0, n}], 3], {n, 0, 61}]; Accumulate[A001285] (* Jean-François Alcover, Sep 25 2012 *)
Join[{0}, Accumulate[1 + ThueMorse /@ Range[0, 100]]] (* Jean-François Alcover, Sep 18 2019, from version 10.2 *)

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n,v[k]=if(k%2,v[k\2+1]-v[k\2])+k\2*3); concat(0,v) \\ Charles R Greathouse IV, May 09 2016

from itertools import accumulate, islice
def A026430_gen(): # generator of terms
    yield from (0,1)
    blist, s = [1], 1
    while True:
        c = [3-d for d in blist]
        blist += c
        yield from (s+d for d in accumulate(c))
        s += sum(c)
A026430_list = list(islice(A026430_gen(),30)) # Chai Wah Wu, Feb 22 2023

def A026430(n): return n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)) # Chai Wah Wu, Mar 01 2023

a(0)=0, a(1)=1, a(2n) = 3n, a(2n+1) = -a(n) + a(n+1) + 3n. - Ralf Stephan, Oct 08 2003

G.f.: x*(3/(1 - x)^2 - Product_{k>=1} (1 - x^(2^k)))/2. - Ilya Gutkovskiy, Apr 03 2019

A360393 Complement of A360392.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 15, 19, 22, 24, 27, 31, 34, 36, 40, 42, 45, 49, 51, 55, 58, 60, 64, 66, 69, 73, 76, 78, 81, 85, 87, 91, 94, 96, 99, 103, 106, 108, 112, 114, 117, 121, 124, 126, 129, 133, 135, 139, 142, 144, 148, 150, 153, 157, 159, 163, 166, 168, 171, 175

Offset: 1

Clark Kimberling, Feb 05 2023

Winston de Greef, Table of n, a(n) for n = 1..5000

Cf. A026430, A360392-A360405.

v = 2 + Accumulate[1 + ThueMorse /@ Range[0, 200]]; (* A360392 *)
Complement[Range[Max[v]], v]    (* A360393 *)

a(n) = if(n < 3, [1, 2][n], 3*n - 5 - hammingweight(n-3)%2) \\ Winston de Greef, Mar 27 2023

from itertools import islice
def A360393_gen(): # generator of terms
    yield from (1,2)
    blist, s = [1], 3
    while True:
        c = [3-d for d in blist]
        blist += c
        for d in c:
            yield from range(s+1,s:=s+d)
A360393_list = list(islice(A360393_gen(),30)) # Chai Wah Wu, Feb 22 2023

A360393(n) = A356133(n-2) + 2 for n>=3

A360394 Intersection of A026430 and A360392.

Original entry on oeis.org

3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 75, 77, 80, 82, 84, 86, 88, 90, 93, 95, 98, 100, 102, 105, 107, 109, 111, 113, 116, 118, 120, 123, 125, 128, 130, 132, 134, 136, 138

Offset: 1

Clark Kimberling, Feb 05 2023

This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively (and likewise for A360402-A360405).
For A360394, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430
u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u
v = u + 1 = A285954, except its initial 1
v' = complement of v.

			(1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
(2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
(3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

Cf. A026430, A356133, A360392, A360393, A356850-A360405.

z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Intersection[u, v]     (* A360394 *)
Intersection[u, v1]    (* A360395 *)
Intersection[u1, v]    (* A360396 *)
Intersection[u1, v1]   (* A360397 *)

A359277 Intersection of A026430 and (1 + A285953).

Original entry on oeis.org

6, 9, 10, 15, 16, 19, 24, 27, 28, 31, 36, 37, 42, 45, 46, 51, 52, 55, 60, 61, 66, 69, 70, 73, 78, 81, 82, 87, 88, 91, 96, 99, 100, 103, 108, 109, 114, 117, 118, 121, 126, 129, 130, 135, 136, 139, 144, 145, 150, 153, 154, 159, 160, 163, 168, 171, 172, 175

Offset: 1

Clark Kimberling, Jan 26 2023

This is the first of three sequences that partition the positive integers. Taking u = A026430 and v = 1 + A285953 (which is A285953 except for its initial 1), the three sequences are (1) u ^ v = intersection of u and v (in increasing order); (2) u ^ v'; and (3) u' ^ v. The limiting density of each of these is 1/3.

			(1)  u ^ v = (6, 9, 10, 15, 16, 19, 24, 27, 28, 31, 36, 37, ...) =    A359277
(2)  u ^ v' = (1, 3, 5, 8, 12, 14, 18, 21, 23, 26, 30, 33, 35, ...) =  A285953, except for the initial 1
(3)  u' ^ v = (2, 4, 7, 11, 13, 17, 20, 22, 25, 29, 32, 34, 38, ...) = A356133

Cf. A026530, A285954, A356133, A359352 to A360139) (results of compositions instead of intersections).

z = 200;
u = Accumulate[1 + ThueMorse /@ Range[0, z]]   (* A026430 *)
u1 = Complement[Range[Max[u]], u]  (* A356133 *)
v = u + 1
v1 = Complement[Range[Max[v]], v]
Intersection[u, v]    (* A359277 *)
Intersection[u, v1]   (* A285953 *)
Intersection[u1, v]   (* A356133 *)

A359352 a(n) = A026430(1 + A026430(n)).

Original entry on oeis.org

3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, 42, 44, 46, 48, 51, 54, 55, 57, 60, 63, 65, 68, 69, 70, 73, 77, 78, 80, 82, 84, 87, 90, 91, 93, 96, 99, 100, 103, 105, 107, 109, 111, 114, 117, 118, 121, 123, 125, 128, 130, 132, 134, 136, 138

Offset: 1

Clark Kimberling, Jan 26 2023

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359353-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v]; (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]     (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]    (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]    (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]   (* A360135 *)

def A359352(n): return (m:=n+1+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)))+(m-1>>1)+(m-1&1|(m.bit_count()&1^1)) # Chai Wah Wu, Mar 01 2023

A360398 a(n) = A026430(1 + A360392(n)).

Original entry on oeis.org

5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, 42, 44, 45, 48, 50, 52, 55, 57, 59, 61, 65, 66, 69, 70, 72, 75, 78, 80, 81, 84, 86, 88, 91, 93, 95, 98, 100, 102, 105, 107, 108, 111, 113, 116, 118, 120, 123, 125, 126, 129, 132, 134, 135, 138

Offset: 1

Clark Kimberling, Feb 10 2023

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360399, A286355, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360402 a(n) = A360392(A026430(n)).

Original entry on oeis.org

3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, 38, 41, 43, 44, 47, 48, 52, 54, 56, 57, 61, 63, 65, 68, 70, 71, 74, 77, 79, 80, 83, 84, 88, 90, 92, 93, 97, 100, 101, 104, 105, 107, 110, 111, 115, 118, 119, 122, 123, 125, 128, 131, 132, 134, 137

Offset: 1

Clark Kimberling, Mar 11 2023

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n) = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively (and likewise for A360394-A360401).

			(1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
(2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
(3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
(4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360403, A360404, A360405.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Table[v[[u[[n]]]], {n, 1, zz}]    (* A360402 *)
Table[v1[[u[[n]]]], {n, 1, zz}]   (* A360403 *)
Table[v[[u1[[n]]]], {n, 1, zz}]   (* A360404 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]  (* A360405 *)

def A360392(n): return n+2+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
def A026430(n): return n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
def A360402(n): return A360392(A026430(n)) # Winston de Greef, Mar 24 2023

A360136 a(n) = 1 + A026430(A026430(n)).

Original entry on oeis.org

2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, 37, 40, 42, 43, 46, 47, 51, 53, 55, 56, 60, 62, 64, 67, 69, 70, 73, 76, 78, 79, 82, 83, 87, 89, 91, 92, 96, 99, 100, 103, 104, 106, 109, 110, 114, 117, 118, 121, 122, 124, 127, 130, 131, 133, 136, 137

Offset: 1

Clark Kimberling, Feb 03 2023

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n)  = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

			(1)  v o u = (2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, ...) = A360136
(2)  v' o u = (1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, ...) = A360137
(3)  v o u' = (4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, ...) = A360138
(4)  v' o u' = (3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, ...) = A360139

Cf. A026530, A359352, A285953, A359277 (intersections instead of results of composition), A359352-A360135, A360137-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];   (* A285953 *)
Table[v[[u[[n]]]], {n, 1, zz}]       (* A360136 *)
Table[v1[[u[[n]]]], {n, 1, zz}]      (* A360137 *)
Table[v[[u1[[n]]]], {n, 1, zz}]      (* A360138 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]     (* A360139 *)

def A360136(n): return 1+(m:=n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)))+(m-1>>1)+(m-1&1|(m.bit_count()&1^1)) # Chai Wah Wu, Mar 01 2023

A359353 a(n) = A026430(A285953(n+1)).

Original entry on oeis.org

1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, 75, 81, 86, 88, 95, 98, 102, 108, 113, 116, 120, 126, 129, 135, 139, 143, 147, 153, 158, 160, 167, 170, 174, 180, 185, 188, 192, 198, 201, 207, 212, 214, 221, 224, 228, 234, 237, 243, 248, 250

Offset: 1

Clark Kimberling, Jan 30 2023

This is the second of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352, A360134-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)

cp's OEIS Frontend

A354384 Difference sequence of A356133.

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A026430 a(n) is the sum of first n terms of A001285 (Thue-Morse sequence).

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A360393 Complement of A360392.

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A360394 Intersection of A026430 and A360392.

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A359277 Intersection of A026430 and (1 + A285953).

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A359352 a(n) = A026430(1 + A026430(n)).

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A360398 a(n) = A026430(1 + A360392(n)).

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A360402 a(n) = A360392(A026430(n)).

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